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Chapter 4: Problem 38

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct mode.) (a) \(\sec 56^{\circ} 8^{\prime} 10^{\prime \prime}\) (b) \(\cos 56^{\circ} 8^{\prime} 10^{\prime \prime}\)

Short Answer

Expert verified
After performing the mentioned steps, we will get the values for secant and cosine given the inputs \(56.1361^{\circ}\). Please use your calculator for specifics.

Step by step solution

01

Conversion to Decimal Degrees

Firstly, convert the given values from degrees, minutes, and seconds to decimal degrees. \nFor example, the decimal equivalent of \(56^{\circ} 8^{\prime} 10^{\prime \prime}\) can be calculated as follows: \n\(56 + \frac{8}{60} + \frac{10}{3600} = 56.1361^{\circ}\).
02

Calculate Secant Function

Now that we have converted our angle to decimal degrees, we can calculate the secant function. Secant is the reciprocal of cosine, so in order to find the secant, we must find the cosine and then take the reciprocal. So, use a calculator to find the cosine of \(56.1361^{\circ}\), then take the reciprocal to find the secant value. Assuming the calculator is in degree mode.
03

Calculate Cosine Function

We can calculate the cosine of the angle using the initial conversion to decimal degrees. Using a calculator in degree mode, find the value for \(\cos(56.1361^{\circ})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Secant Function
The secant function is a fascinating trigonometric function often used in various mathematical applications. The secant of an angle \( \theta \), written as \( \sec(\theta) \), is defined as the reciprocal of the cosine function. This means it can be expressed as:
  • \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
Understanding this relation helps when working with angles, particularly when calculations use the secant function in real-world problems. By using a calculator, one can find the \( \cos(\theta) \), then simply divide 1 by this value to obtain the secant. Keep in mind, proper calculator settings, such as the mode, ensure correct evaluations.
Cosine Function
The cosine function is a core trigonometric function vital for various calculations. It deals with the adjacent side over the hypotenuse in a right-angled triangle. The angle in question here, \( 56.1361^{\circ} \), is a direct result of converting from degrees, minutes, and seconds.The formula for converting between these units inherently involves cosine calculations. Using a calculator, you can directly compute \( \cos(56.1361^{\circ}) \) to find the precise value. Remember, different units require the calculator to be in a specific mode, such as degree mode, to ensure accuracy in cosine evaluations.
Degree Mode
In trigonometry, degree mode is crucial for correctly evaluating trigonometric functions when angles are measured in degrees. When you perform any calculation involving degrees, it鈥檚 essential to set your calculator to degree mode. How can you identify if this mode is active? It usually displays 'DEG' or something similar on its screen. To ensure accuracy:
  • Check the calculator's current mode before calculations.
  • Switch to degree mode if necessary.
  • Ensure all angle inputs are in degrees and not radians.
Maintaining the right mode prevents errors in trigonometric calculations, ensuring that functions like cosine or secant yield precise results.
Conversion to Decimal Degrees
Converting from degrees, minutes, and seconds (DMS) to decimal degrees is fundamental when dealing with trigonometric calculations. It's essential to understand this conversion as it simplifies the angle's representation, making it usable in calculations.To convert from DMS to decimal degrees:
  • Take the number of degrees.
  • Add the minutes divided by 60.
  • Add the seconds divided by 3600.
For example, to convert \( 56^{\circ} 8^{\prime} 10^{\prime \prime} \):\[ 56 + \frac{8}{60} + \frac{10}{3600} = 56.1361^{\circ} \]This process is crucial for ensuring the angle is ready for input into calculators and software using decimal degrees instead of DMS format.

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Most popular questions from this chapter

A company that produces snowboards forecasts monthly sales over the next 2 years to be $$S=23.1+0.442 t+4.3 \cos \frac{\pi t}{6}$$ where \(S\) is measured in thousands of units and \(t\) is the time in months, with \(t=1\) representing January 2014 Predict sales for each of the following months. (a) February 2014 (b) February 2015 (c) June 2014 (d) June 2015

The displacement from equilibrium of an oscillating weight suspended by a spring is given by \(y(t)=2 \cos 6 t,\) where \(y\) is the displacement (in centimeters) and \(t\) is the time (in seconds). Find the displacement when (a) \(t=0,\) (b) \(t=\frac{1}{4}\) and \((\mathrm{c}) t=\frac{1}{2}.\)

Define the inverse cosecant function by restricting the domain of the cosecant function to the intervals \([-\pi / 2,0)\) and \((0, \pi / 2],\) and sketch the graph of the inverse trigonometric function.

The numbers of hours \(H\) of daylight in Denver, Colorado, on the 15 th of each month are: \(1(9.67), 2(10.72), 3(11.92), 4(13.25)\) \(5(14.37), \quad 6(14.97), \quad 7(14.72), \quad 8(13.77), \quad 9(12.48)\) \(10(11.18), \quad 11(10.00), \quad 12(9.38) . \quad\) The month is represented by \(t,\) with \(t=1\) corresponding to January. A model for the data is $$H(t)=12.13+2.77 \sin \left(\frac{\pi t}{6}-1.60\right)$$. (a) Use a graphing utility to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the context of the problem? Explain.

Distance A plane flying at an altitude of 7 miles above a radar antenna will pass directly over the radar antenna (see figure). Let \(d\) be the ground distance from the antenna to the point directly under the plane and let \(x\) be the angle of elevation to the plane from the antenna. (d is positive as the plane approaches the antenna.) Write \(d\) as a function of \(x\) and graph the function over the interval \(0 < x < \pi\).
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